Short CommunicationA comparison of linear reformulations for multinomial logit choice probabilities in facility location models
Introduction
When customer choice behavior is considered in facility location planning we generally assume utility maximizing behavior. Probabilistic demand based on utility maximization implies that only the probability of a customer choosing a given facility is known. Traditionally, in facility location literature, gravity-type demand models are used to consider probabilistic demand (Serra, Eiselt, Laporte, & ReVelle, 1999). More recently, there is a growing body of literature that considers the multinomial logit model (MNL) in facility location models (see for example Aros-Vera et al., 2013, Benati and Hansen, 2002, Haase, 2009, Haase and Müller, 2013, Müller et al., 2009, Zhang et al., 2012). The MNL is a well-known discrete choice model (random utility model) to describe (spatial) customer choice behavior (Ben-Akiva et al., 2002, McFadden, 2001, Müller et al., 2012). Using MNL within a mathematical program for locational decision making probably yields a non-linear model formulation (see Benati, 1999, Marianov et al., 2008, for example). If we consider only locational decisions, then there exist three different linear reformulations of the MNL yielding mixed-integer linear programs. Benati and Hansen (2002) where the first who proposed a linear reformulation based on variable substitution. Haase (2009) has proposed to employ the constant substitution pattern of the MNL in order to enable a linear integer formulation (see also Aros-Vera et al., 2013). Finally, Zhang et al. (2012) presented an alternative approach based on variable substitution.
As the three model formulations are discussed independently so far, we compare them in this contribution. Therefore, we first give a brief discussion of the MNL and its incorporation into the maximum capture problem (Section 2). Based on this, we discuss the three linear reformulations using a unified notation in order to make them more comparable (Sections , , ). In Section 3 we compare the models in a numerical study using synthetic data. Our conclusions can be found in Section 4.
Section snippets
Mathematical formulations
Consider a firm that wishes to enter a market where customers are located in zones denoted by nodes I. Potential facilities of the firm and facilities of competitors are located in nodes M. Now, the problem of the firm is to select r facilities from all potential facilities J with J ⊂ M such that the patronage of the facilities of the firm is maximized. In order to determine the patronage of a located facility we assume the customers located in i ∈ I to be homogeneous in their observable
Numerical investigations
We compare the models using artificially generated data. Therefore, we have implemented the three models in GAMS (23.7) and use CPLEX 12 on a 64-bit Windows Server 2008 with 1 Intel Xeon 2.4 gigahertz processor and 24 gigabyte RAM to solve the problems. The Cartesian coordinates of the nodes I and M are randomly generated by a uniform distribution in the interval [0, 30]. We consider the rectangular distances between i ∈ I and j ∈ M denoted by dij. The deterministic part of utility of (1) is defined
Conclusion
In the last decade three linear reformulations for MNL-based demand in facility location models have been proposed. All references appear to be independent from each other. In order to compare the model formulations in terms of solvability we first present the models using a unified notation. This is followed by a computational study using artificial data. We find that the approach proposed by Haase (2009) seems to be promising for solving large problems using GAMS/CPLEX. This finding has
Acknowledgments
The authors gratefully thank an anonymous referee for helpful comments on an earlier version of the paper.
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